Momentum Conservation and Collisions

Overview

When objects interact over a short time interval, such as:

• collisions
• explosions
• recoil
• separation of connected bodies

the internal forces between them are often large, but these forces occur in equal and opposite pairs. As a result, the total momentum of the system can remain constant.

This makes momentum conservation one of the most powerful tools in H2 Physics.

Related hub: Dynamics

Why It Matters

Momentum conservation often simplifies violent or short interactions where direct force analysis would be difficult.

Definition

A system conserves momentum if the resultant external force is zero or the external impulse is negligible during the interaction.

Key Representations

$$\sum {\vec{p}}_{\text{initial}}=\sum {\vec{p}}_{\text{final}}$$ $$m_1{u}_1+m_2{u}_2=m_1{v}_1+m_2{v}_2$$ $$u_1-u_2=v_2-v_1$$

Isolated Systems

Definition

A system is isolated if the resultant external force on it is zero or negligible during the interaction.

$$\sum {\vec{F}}_{\text{external}}=0$$

Then:

$$\text{Total momentum before}=\text{Total momentum after}$$

Why It Works

Internal forces occur in action-reaction pairs:

• equal in magnitude
• opposite in direction

These cancel within the system, so only external forces can change total momentum.

Examples of Approximate Isolation

During short collisions:

• weight acts, but short collision time means external impulse is small
• ground reactions may be negligible in chosen direction

Hence momentum conservation is often valid horizontally.

Principle of Conservation of Momentum

For two bodies in one dimension:

$$m_1{u}_1+m_2{u}_2=m_1{v}_1+m_2{v}_2$$

Where:

• $u$ = initial velocity
• $v$ = final velocity

Use signs according to chosen positive direction.

Choosing Sign Convention

Example: Right positive.

• motion right → positive
• motion left → negative

Then substitute velocities with signs.

This avoids directional confusion.

Collisions

A collision is any interaction where bodies exert large forces on each other for a short time.

Examples:

• billiard balls
• carts on track
• car crash
• bat and ball

Momentum is conserved if external forces are negligible.

Types of Collisions

Elastic Collision

Conserved:

• momentum
• kinetic energy

For 1D head-on collisions:

$$u_1-u_2=v_2-v_1$$

(relative speed of approach = relative speed of separation)

---

Inelastic Collision

Conserved:

• momentum only

Kinetic energy decreases.

Energy is transferred to:

• heat
• sound
• deformation
• internal energy

---

Perfectly Inelastic Collision

Bodies stick together after collision.

Hence:

$$v_1=v_2=V$$

Use common final velocity.

Recoil

When one object ejects another, momentum is conserved.

Example:

• gun and bullet
• cannon and shell
• astronaut throwing tool

If initial momentum is zero:

$$m_1{v}_1+m_2{v}_2=0$$

Thus bodies move in opposite directions.

Explosions

An object initially at rest breaks into fragments.

Initial momentum:

$$0$$

Therefore vector sum of final momenta must also be zero.

Worked Examples

Example 1: Two Trolleys Collide

A 2.0 kg trolley moves right at 4.0 m s${}^{-1}$ and collides with a stationary 3.0 kg trolley. They move together.

Use conservation of momentum:

$$2(4)+3(0)=(2+3)V$$ $$8=5V$$ $$V=1.6{\text{m s}}^{-1}$$

to the right.

This is perfectly inelastic.

---

Example 2: Recoil of Gun

A 0.020 kg bullet moves right at 300 m s${}^{-1}$. Gun mass is 3.0 kg.

Initially at rest:

$$0=3v+0.020(300)$$ $$0=3v+6$$ $$v=-2.0{\text{m s}}^{-1}$$

Gun recoils left.

---

Example 3: Elastic Collision

1.0 kg mass moving right at 6.0 m s${}^{-1}$ collides head-on with stationary 1.0 kg mass.

For equal masses in elastic collision:

They exchange velocities.

Final velocities:

• first mass: 0
• second mass: +6.0 m s${}^{-1}$

---

Example 4: Opposite Directions

Take right positive.

A 2.0 kg body moves right at 5.0 m s${}^{-1}$.
A 1.0 kg body moves left at 3.0 m s${}^{-1}$.
After collision they stick.

$$2(5)+1(-3)=3V$$ $$10-3=3V$$ $$V=\frac{7}{3}=2.33{\text{m s}}^{-1}$$

right.

Relative Speed Condition (Elastic, 1D)

For head-on elastic collisions:

$$u_1-u_2=v_2-v_1$$

Use together with momentum conservation.

This gives two equations for two unknowns.

Special Cases

Equal Masses, One Initially at Rest

Elastic collision:

Velocities exchanged.

Very Massive Target

Small object rebounds with nearly same speed.

Example:

ball striking wall.

Perfectly Inelastic

Maximum kinetic energy loss consistent with momentum conservation.

Momentum vs Kinetic Energy

Momentum can be conserved even when kinetic energy is not.

This is common in real collisions.

Do not assume kinetic energy conservation unless collision is stated elastic.

Common Exam Pitfalls

1. Ignoring External Forces

Check whether system can be treated as isolated.

2. Wrong Signs

Opposite directions require negative velocities.

3. Assuming KE Conserved in Every Collision

False unless elastic.

4. Forgetting Common Final Velocity

If objects stick together:

$$v_1=v_2$$

5. Mixing Scalars and Vectors

Momentum has direction.

6. Using Relative Speed Formula for Inelastic Collision

Only valid for elastic collisions.

Problem-Solving Strategy

For Collision Questions

1. Choose positive direction.
2. Write momentum conservation equation.
3. Add second condition if needed:
   • kinetic energy conserved, or
   • common final velocity, or
   • relative speed condition
4. Solve algebra carefully.
5. Check sign and physical meaning.

Summary Table

Interaction	Momentum Conserved?	KE Conserved?
Elastic collision	Yes	Yes
Inelastic collision	Yes	No
Perfectly inelastic	Yes	No
Explosion / recoil	Yes	Not generally

Formula Summary

$$m_1{u}_1+m_2{u}_2=m_1{v}_1+m_2{v}_2$$ $$u_1-u_2=v_2-v_1$$ $$v_1=v_2=V\text{(stick together)}$$

If initially at rest:

$$\sum {\vec{p}}_{final}=0$$

Summary

Momentum conservation is most powerful when the interaction is brief and the external impulse on the chosen system is negligible.

Links

Related Links

• Dynamics
• Momentum and Impulse
• Free-Body Diagrams and Force Analysis
• Forces
• Kinematics
• Work, Energy and Power

$$\text{relative speed of approach}=\text{relative speed of separation}$$

This condition is not valid for inelastic collisions.

Common Exam Points

• Momentum conservation requires a clearly defined system.
• Kinetic energy is conserved only for elastic collisions.
• Momentum is vectorial, so signs and directions matter. In one dimension, choose a positive direction and use signed components.
• Objects sticking together after collision means the collision is perfectly inelastic.
• The relative velocity condition applies only to one-dimensional elastic collisions.

Links

• Related: dynamics
• Related: conservation laws in physics
• Related: work energy and power
• Related: momentum and impulse
• Related: dynamics methods and non constant forces
• Misconception: momentum conservation system
• Misconception: inelastic ke conservation
• Misconception: elastic relative speed application